Uso de modelos não-lineares para o ajuste de curvas de crescimento de cavalos pantaneiros

USING NONLINEAR MODELS TO DESCRIBE HEIGHT

GROWTH CURVES IN PANTANEIRO HORSES

1

SANDRA APARECIDA SANTOS2_{, GERALDO DA SILVA E SOUZA}3_, MARCOS RUBEN DE OLIVEIRA4 _{and JOSÉ ROBSON SERENO}5

ABSTRACT - Height at withers data mostly from birth to 36 months of age of 26 Pantaneiro horses were used to fit Brody, Richards, Gompertz, Logistic, Weibull and Morgan-Mercer-Flodin nonlinear response functions. Based on measures of average curvature and combined mean square error, the Weibull model was chosen. The asymptote of this curve, representing the average height at maturity, was higher for males than females. The maturity index, however, was more elevated for females than males. There was indication of a negative association between the maturity index and height at matu-rity only for females. These results indicate that females mature earlier. After checking for normality and homogeneity of variances within groups (sex) the analysis investigated sex differences via t-tests. A significant difference was detected only for height at birth.

Index terms: curvature, growth, height-age relationships.

USO DE MODELOS NÃO-LINEARES PARA O AJUSTE DE CURVAS DE CRESCIMENTO DE CAVALOS PANTANEIROS

RESUMO - Dados de altura da cernelha e idade de 26 cavalos Pantaneiros, obtidos, na maioria, do nascimento até 36 meses, foram ajustados aos modelos de respostas não-lineares de Brody, Richards, Gompertz, Logístico, Weibull e Morgan-Mercer-Flodin. Estes seis modelos matemáticos foram com-parados com o uso de uma medida de curvatura média e do erro médio quadrático combinado. O modelo de Weibull foi escolhido. A assíntota desta curva representa a altura esperada na maturidade. Os machos apresentaram um valor maior desta quantidade do que as fêmeas. O índice de maturidade, contudo, é maior para as fêmeas. Observou-se uma indicação de associação negativa entre altura na maturidade e índice de maturidade somente nas fêmeas. Tais resultados indicam que as fêmeas amadu-recem mais cedo. Após testes de normalidade e homogeneidade de variância, as diferenças entre sexos foram analisadas com o uso do teste-t. Somente a altura da cernelha ao nascimento apresentou diferen-ça significativa.

Termos para indexação: curvatura, crescimento, relações altura-idade.

(Robinson, 1976). Hyperplasy (an increase in cell

number) and hypertrophy (an increase in cell size)

determine to what extent an animal increases in

weight and size, and development determines the

shape of an animal along with various organ

func-tions. The study of animal growth is often described

as the integration of all aspects of animal science

including nutrition, genetics or animal breeding,

physiology and meat science (Trenkle & Marple,

1983).

Growth curves reflect the lifetime

interrelation-ships between an individual’s inherent impulse to

grow and mature all body parts and the environment

in which these impulses are expressed. Knowledge

of the growth curve is important to all animal

scien-1_{Accepted for publication on February 18, 1999.}

2_{Zootecnist, M.Sc., Embrapa-Centro de Pesquisa Agropecuária}

do Pantanal (CPAP), Caixa Postal 109, CEP 79320-900 Corumbá, MS, Brazil. E-mail: sasantos@cpap.embrapa.br

3_{Statistician, Ph.D., Embrapa-Secretaria de Administração}

Estratégica (SEA), CEP 70770-901 Brasília, DF, Brazil.

4_{Statistician, Dep. de Estatística, Universidade de Brasília,}

CEP 70910-900 Brasília, DF, Brazil.

5_{Veterinarian, M.Sc., Embrapa-CPAP.}

INTRODUCTION

Populations can be characterized by their growth

patterns but growth is a complex biological

phenom-enon with no adequately defined direct measure

tists, regardless of specialization, who are concerned

with the effects of their research and

recommenda-tions on lifetime production efficiency (Fitzhugh,

1976). Growth curves have been characterized by

plotting measures of size (weight, height, width)

against age. The use of regression models to describe

growth condenses the information contained in a

time series into a few biologically interpretable

pa-rameters (Eisen, 1974). Several models are

avail-able for this purpose and experimental comparisons

are needed to assist in the choice of the most

appro-priate model (Brown et al., 1976). Mathematical

nonlinear models to describe growth have been

de-veloped by Brody (1945), Richards (1959), Nelder

(1961), and Ratkowsky (1983). These models have

been used in beef cattle to describe weight-age

rela-tionships (Brown et al., 1972, 1976; Duarte, 1975;

De Nise & Brinks, 1985; Nobre et al., 1987) but

have not been used in horses.

The majority of the studies on horse growth are

based on weight, height, chest girth and cannon bone

(Green, 1961, 1969, 1976; Hintz et al., 1979; Santos,

1989). Of these measurements, height has more

prac-tical interest because it is required for the purpose

of description and classification of horses, while

weight is more a measure of the physical condition

of the animal (Reed & Dunn, 1977; Hickman &

Collis, 1984). According to Trenkle & Marple (1983)

growth curves can be of significant value in future

selection programs and in the study of the

energet-ics of growth.

The purpose of the present study was to choose,

among alternative nonlinear response functions, the

best model in regard to goodness of fit,

computa-tional ease, and validity of large sample

approxima-tions in small samples. The sex and age effects on

the growth of Pantaneiro horses in the Brazilian

Pantanal were also evaluated.

MATERIAL AND METHODS

The data on measurements of height at withers were obtained for 14 males and 12 females at Nhumirim farm, Nhecolândia sub-region, Brazilian Pantanal, during the period from 1990 to 1995. Monthly height at withers measures were taken until 3 months of age and at 6, 12, 18, 24, and 36 months of age. Eight animals were also

measured at 48 months of age. Most measurements were taken by the same person, however variation can occur due mainly to the levelness of the ground. Height at with-ers is a vertical distance from the ground to the highest protruding thoracic vertebrae. The measurements were taken with a ‘hipometer’. Only horses with 1080 or more days of age were included in the analyses. Breeding sea-sons lasted last four months (November to February) and three sires were used in this period. The animals were maintained in native pasture and they received common salt ad libitum. A growth curve was fit to each animal. Six nonlinear response functions were selected for curve fitting: 1. Brody growth function: Ht = a(1-be-kt); 2.

Richards growth function: Ht = a(1-be-kt)m; 3. Gompertz

growth function: Ht = a exp{-exp(b-kt)}; 4. Logistic

growth function: t _{b kt} e 1 a H ₋ +

= ; 5. Weibull growth func-tion: _{= −} −ktδ t a be H ; 6. Morgan-Mercer-Flodin - MMF growth function: _δδ + + = t k at bk

Ht . These models are

de-scribed in detail in Ratkowsky (1983) where one may also find handy techniques to obtain initial parameter estimates in each case. They are all used to fit the same type of data. The ordinary least squares estimates, for each animal and each model, were computed using the modified Gauss-Newton method of Hartley (1961) and PROC NLIN of SAS (1990). In regard to these fits five animals (four males and one female) showed atypical behavior and were ex-cluded from further analyses. In this context the number of animals under investigation was reduced to 21 (10 males and 11 females). For the excluded horses in all instances but one either convergence was not obtained or it was very slow for some of the models. In the exception case convergence results were sound but the parameter esti-mates were far different (outliers) suggesting the presence of a distinct population. The model MMF did not con-verge for any of the animals. This is the only instance were we can report a serious computational difficulty.

A common characteristic of all growth models above is that they share at least two biologically relevant param-eters. The asymptote a representing average size at matu-rity and the parameter k. The latter is a function of the ratio of the maximum growth rate to mature size. It is commonly referred as maturity index. For the Weibull model, the parameter δ δ δ δ δ can also be seen as a maturity

in-dex. The smaller and positive are its values the slower will be the approximation of the corresponding growth curves to their respective asymptotes. We attach no gen-eral special meaning to band to m, the latterin the case of the Richards response. They both influence the speed at which the asymptote is attained. With a positive associa-tion between band k, a large maturity index will imply a large b and the combined effect will be a fast maturity rate. Another parametric function of biological importance is the height at birth, which is given by a(1-b), a(1-b)m_,

a exp {-eb_{}, a/(1+e}b_{) and a-b, respectively, for models 1,}

2, 3, 4, 5 e 6.

Two diagnostic measures were used to choose among the alternative response functions. A measure of nonlinearity (intrinsic curvature) and a measure of good-ness of fit. Curvature measures have been introduced in the statistical literature by Beale (1960) and extended by Bates & Watts (1980). The concept is extensively used by Ratkowsky (1983) to choose among alternative models and to assess the validity of large sample statistical proce-dures in small samples. It is a diagnostic tool intended to measure departures from linearity. The more linear a model is the more likely it is to show desirable computational and distributional properties for nonlinear least square estimates. Small curvatures imply that biases in the esti-mation of parameters and their variances will be negli-gible and the corresponding statistical distributions will be close to normal. Curvatures appear intwo kinds: in-trinsic curvature, a property of the response function in use, and parametric curvature, a property of the particular parametrization used in the definition of the response func-tion. A large parametric curvature may be reduced by a suitable reparametrization of the model function. A large intrinsic curvature, on the other hand, may put in jeop-ardy the use of asymptotic theory in a particular applica-tion. The concepts are presented in full detail in Seber & Wild (1989) were one may also find the computing algo-rithm suggested by Bates & Watts (1980). Oliveira (1996) developed a SAS macro to implement the Bates and Watts procedure which is used in this study to compute intrinsic curvatures.

The measure of goodness of fit used was the combined mean squares error (MSE).

For model i, i j N 1 j i j N 1 j i

df

SSE

MSE

= =

∑

∑

=

where N is the number of animals,

_SSE

i_j is the residual sum of squares resulting from the fit of model i to the jth animal, and

df

_ji is the corresponding error degrees of freedom.

The presence of sex and age differences was investi-gated via t-tests on the parameters, exploring the replica-tion pattern imbedded in the experiment.

RESULTS AND DISCUSSION

Table 1 shows the average value of the intrinsic

curvature and the combined mean square error for

each of the nonlinear response functions considered

in this study. The two models with best prediction

power (smallest combined mean square error) are

Richards and Weibull. Between these, the Weibull

model showed a considerable lower level of

nonlinearity. It is worth to mention that the

coeffi-cients of determination for the Weibull fits range

from 0.988 to 0.999. The average value for females

is 0.994 and for males 0.993. Since the Richards

response function has an essentially equivalent

pre-diction power, similar results apply for the

corre-sponding model.

Table 2 shows average values of parameter

esti-mates for the Weibull model for each sex.

Confi-dence intervals were evaluated under the

assump-tion that parameter estimates represent samples from

normal populations centered at the corresponding

true population values. This approach is convenient

to deal with the correlation problem imbedded in

the study of growth curves. Although observations

taken on the same individual may be correlated,

pa-TABLE 1. Intrinsic curvature and combined mean square error for growth models according to sex.

Model Sex Intrinsic

curvature mean squareCombined Brody Female 0.12619 4.05749 Male 0.11718 4.43629 Richards Female 0.58829 2.47668 Male 1.70630 3.02947 Gompertz Female 0.01045 5.18596 Male 0.01006 5.50570 Logistic Female 0.15674 5.50570 Male 0.14871 6.01068 Weibull Female 0.16394 2.50996 Male 0.18146 3.17833

Parameter Sex Average Lower confidence limit Upper confidence limit a Female 139.016 135.229 142.802 Male 140.225 136.830 143.621 b Female 58.913 52.534 65.292 Male 52.931 48.133 57.729 log(k) Female 0.372 0.245 0.498 Male 0.299 0.188 0.409 k Female 1.472 1.278 1.645 Male 1.360 1.207 1.505 δ Female 0.732 0.629 0.835 Male 0.835 0.776 0.894 a-b Female 80.103 74.068 86.137 Male 87.294 84.493 90.096

TABLE 2. Average values and 95% confidence intervals for parameters of the Weibull model.

TABLE 3. Values of t-tests for parameters of interest, Weibull model. The method of Satterthwaite was used for a-b (SAS, 1990). Parameter Df t-value p-value

a-b 14 -2.5668 0.0224

a 19 -0.5610 0.5813

b 19 1.7523 0.0958

δ 19 -2.0067 0.0592

log(k) 19 1.0244 0.3185

FIG. 1. Height model functions for males and females based on average values of parameters obtained with the Weibull model.

rameters, obtained for different individuals, inherit

the independence of the sampling scheme. Only the

maturity index

k

fails to pass Shapiro-Wilks

nor-mality test (SAS, 1990). However log(

k

) has a

dis-tribution that can be fairly closely approximated by

a normal. In this context it was computed a

confi-dence interval for log(

k

). The interval was inverted

to set up confidence bounds for

k

.

The asymptote

a

, the average height at maturity,

was only 1.2 cm higher for males than for females

(Table 2). The maturity index

k

was larger for

fe-males than fe-males (1.472 and 1.360, respectively).

The differences were not markedly significant since,

as shown in Table 3, the corresponding hypothesis

of equality cannot be rejected. The Weibull curves

for each sex, based on the average values of

Table 2, are shown in Fig. 1.

Only height at birth fails the test of homogeneity

of variances.

Table 3 reports the findings. Only a-b (height at

birth) showed a real sex difference. Females had a

lesser height at birth than males. This did not agree

with the findings of Reed & Dunn (1977) for the

Arabian horse. The parameters

δδδδδ

and

b

showed

mar-ginal significance. Females showed a smaller value

of

δδδδδ

and a larger value of

b

.

The relationship between asymptotic growth level

and maturity may be assessed from Table 4 where

Height (cm) 140 130 120 110 100 90 80 70 0 1 2 3 4 5 Age (years)

E(h) Males = 140.225 - 52.931 exp (-1.360t.835₎

TABLE 4. Spearman (rank) correlation coefficients obtained by the Weibull model. Values in parenthesis are p-values for the null hypothesis that the coefficient is zero.

Parameter Male Female

a b δ k a b δ k a 1.00 0.75 -0.10 -0.07 1.00 0.44 -0.62 -0.57 (0.01) (0.78) (0.85) (0.18) (0.04) (0.07) b 1.00 -0.55 -0.08 1.00 -0.79 0.31 (0.09) (0.83) (0.00) (0.36) δ 1.00 0.12 1.00 -0.10 (0.75) (0.77) k 1.00 1.00

Spearman (rank) correlation coefficients are shown.

There is a negative association between

a

and

k

(and

δδδδδ

)

only for females.

The larger maturity index estimate and the

nega-tive association between

a

and

k

observed for the

female group indicate that females mature earlier

than males. This is confirmed by a smaller value of

δδδδδ

(and a larger

b

)

for the female group and the

sig-nificant negative association of this parameter with

the asymptote. A further indication the females

ma-ture earlier, following De Nise & Brinks (1985), is

the significant smaller height at birth for females and

the positive association between

b

and

k

for the

group. According to Brown et al. (1972) different

mature size may or may not represent different

pat-terns of growth. In the present study, the mature

height obtained for males was bellow the average

(142 cm ) obtained for Pantaneiro males registered

in the Association ABCCP. For females the

oppo-site was observed (the Association average is

137 cm ). Both values are within the confidence

lim-its of Table 3 but the value for males is quite close

to the upper bound. There are two explanations for

this. Firstly the males registered with the

associa-tion are selected animals and secondly the males

observed in the present study may not have reached

maturity. Reed & Dunn (1977) studying Arabian

horses observed that the mature height at withers

for females was achieved by 48 months of age

whereas males grew another 1.0 cm from 48 to 60

months of age. Therefore, in order to have a good fit

to asymptotic height, 60 months of age is necessary

for males. In this context is also important to

em-phasize that the animals we observed were

main-tained in native pastures without a feed supplement.

CONCLUSIONS

1. The Weibull response function is the best

para-metric model to fit height-age data for Pantaneiro

horses.

2. The male response curve slightly dominates.

ACKNOWLEDGEMENTS

To Maria Cristina Medeiros Mazza; for the

as-sistance to Roberto Rondon and Márcio Silva and

others; for the collaboration in the collection of the

data.

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